Issue 59

T.-H. Nguyen et alii, Frattura ed Integrità Strutturale, 59 (2022) 172-187; DOI: 10.3221/IGF-ESIS.59.13

Figure 4: Convergence curves of the AdaBoost-DE and the DE.

Tab. 5 shows that the optimal results of the AdaBoost-DE, the DE, the aeDE, and the mSOS are the same and better than the RBAS (1336.634 kg for the AdaBoost-DE, the DE, the aeDE, the mSOS, and 1336.7493 kg for the RBAS). The optimal weights found by the MMSM (1329.715 kg) [5] and by the PSOC (1327.048 kg) [35] are smaller than the optimal weight found in the present work. However, according to our verification, these designs slightly violate the buckling constraint at elements 52 and 80. Thus, they are not reported in Tab. 5. Despite having the same optimal results, the advantage of the proposed method is the speed. The AdaBoost-DE conducts 22488 structural analyses while the required number of structural analyses of the RBAS, the DE, the aeDE, the mSOS are 90000 times, 37500 times, 23925 times, and 24000 times, respectively. Fig. 4 clearly exhibits the benefit of integrating the AdaBoost classification technique into the DE algorithm. During 750 iterations, the DE requires 37500 analyses while the AdaBoost-DE performs only 22488 analyses. In other words, the application of the AdaBoost technique reduces 40% of structural analyses. Compared with the results in Ref. [30], the modifications proposed in the present work increase the reduction rate from 20% to 40%. Regarding the computing time, although the proposed method requires additional time for training and employing the machine learning model, it is still more time-efficient than the original DE algorithm. According to Tab. 5, the average time consumed by the DE and the AdaBoost-DE are 3036.6 s and 2024.5 s, respectively. It means the AdaBoost-DE is approximately 1.5 times faster than the DE in terms of computing time. Validation of the optimal design using the commercial software In this section, the best solution found by the AdaBoost-DE is re-checked using the commercial software CSI SAP2000. This is a well-known software that has been widely utilized by structural engineers over the world. The section properties including sectional areas as well as the radius of gyration of 38 member groups of the best solution found by the AdaBoost- DE are reported in Tab. 6. A 3D model containing 52 nodes and 160 elements is simulated by using SAP2000 as schematized in Fig. 5(a). The material steel used in this model is specified with the following properties: the modulus of elasticity E =2.047E+06 kgf/cm 2 and the weight per unit volume  =7.85E-03 kg/cm 3 . Elements of the model are assigned to 38 different sections with the section properties as presented in Tab. 6. Eight static load cases are defined where the force magnitudes of these load cases are taken from Tab. 3. The model is linearly analyzed and the stress contour of the envelope load combination is plotted in Fig. 5(b). Next, the tensile stress and buckling stress constraints are checked based on the obtained results. The maximum tensile stress found at Element 86 reaches 912.1 kgf/cm 2 , which equals 61% of the limitation. The critical element according to the buckling stress condition is Element 105. The compression stress in this element is equal to -386.3 kgf/cm 2 , achieving 100% of the allowable buckling stress. Thus, the optimal design does not violate any design constraints. Furthermore, the stresses of 160 elements obtaining by SAP2000 and by in-house FEA code are compared in Fig. 6. The coefficient of determination R 2 =0.9996 is very close to 1.0 indicating the high similarity of SAP2000 and the developed FEA code.

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